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<div id="Sx1" class="ltx_section">
<h1 class="ltx_title ltx_title_section">Generator of simulated data</h1>

<div id="Sx1.p1" class="ltx_para">
<p class="ltx_p">ThunderSTORM is capable of generating a sequence of SMLM-like images
in which the ground-truth positions of the molecules are known. This
allows users to perform Monte Carlo simulations <cite class="ltx_cite">[<a href="#bib.bib38" title="Real-time computation of subdiffraction-resolution fluorescence images" class="ltx_ref">4</a>, <a href="#bib.bib17" title="Minimizing detection errors in single molecule localization microscopy" class="ltx_ref">3</a>]</cite>
and to quantitatively evaluate the performance of applied localization
algorithms by calculating, e.g., the Jaccard index or <img id="Sx1.p1.m1" class="ltx_Math" style="vertical-align:-5px" src="mi/mi9.png" width="23" height="18" alt="F_{1}"> score
using the <a href="PerformanceEvaluationPlugIn.html" title="" class="ltx_ref">Performance evaluation plugin</a>.
In addition to the image size and sequence length, users can specify
the intensity, imaged size, and spatial density of the generated molecules.
The resulting images can be subjected to sample drift. Noise in the
generated images simulates the behavior of CCD or EMCCD cameras.</p>
</div>
<div id="Sx1.SSx1" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">Image formation</h2>

<div id="Sx1.SSx1.p1" class="ltx_para">
<p class="ltx_p">For each frame, we first create an ideal, noise free, SMLM-like image
to simulate the expected number of photons detected in each camera
pixel. Image formation starts by creating a list of molecules with
FWHM and intensity chosen randomly in user-specified ranges, and with
random positions of molecules given by a user-specified spatial density
(see below). Users can also specify any of the implemented PSF models,
including 3D models, to create the simulated images of molecules.
The generated molecules are added sequentially to the final image
similarly as in the <a href="rendering/ui/DensityRenderingUI.html" title="" class="ltx_ref">Gaussian rendering</a>
method. A user-specified offset is added to the generated image sequence
to simulate photon background. Alternatively, a gray-scale image,
in which each pixel value is normalized to the interval <img id="Sx1.SSx1.p1.m1" class="ltx_Math" style="vertical-align:-6px" src="mi/mi13.png" width="39" height="21" alt="\left[0,1\right]">,
can be used as a weighting factor of the offset level in different
parts of the generated images to simulate an irregular background
as might be encountered in real samples.</p>
</div>
<div id="Sx1.SSx1.p2" class="ltx_para">
<p class="ltx_p">In order to simulate the photon counting process in the generated
images, each pixel value expressed in photons is modified by a Poisson-distributed
random number. The data generator can optionally simulate EM gain
of EMCCD cameras. In this case we use a stochastic model described
in <cite class="ltx_cite">[<a href="#bib.bib11" title="A stochastic model for electron multiplication charge-coupled devices–from theory to practice." class="ltx_ref">1</a>]</cite>, where the electron multiplication is modeled
by the Gamma distribution <img id="Sx1.SSx1.p2.m1" class="ltx_Math" style="vertical-align:-6px" src="mi/mi11.png" width="55" height="21" alt="\Gamma(k,g)">. The shape <img id="Sx1.SSx1.p2.m2" class="ltx_Math" style="vertical-align:-2px" src="mi/mi17.png" width="14" height="16" alt="k"> is given
by the number of photons detected in the simulated CCD register and
the scale <img id="Sx1.SSx1.p2.m3" class="ltx_Math" style="vertical-align:-5px" src="mi/mi5.png" width="13" height="15" alt="g"> is given by the user-specified value of gain.</p>
</div>
<div id="Sx1.SSx1.p3" class="ltx_para">
<p class="ltx_p">Finally, the signal in the camera register is digitized by converting
the photons to digital counts. The CCD sensitivity (in photons per
A/D count), and the camera digitizer offset (in A/D counts) are user-specified
in the <a href="CameraSetupPlugIn.html" title="" class="ltx_ref">camera setup</a>, as well as the
camera pixel size (in nanometers) as projected to the sample plane.</p>
</div>
</div>
<div id="Sx1.SSx2" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">Fixed or spatially varying density of molecules</h2>

<div id="Sx1.SSx2.p1" class="ltx_para">
<p class="ltx_p">Users can specify a fixed or spatially varying density of simulated
molecules in the generated images. The variability is achieved by
a user-supplied gray-scale mask <img id="Sx1.SSx2.p1.m1" class="ltx_Math" style="vertical-align:-2px" src="mi/mi10.png" width="23" height="16" alt="M">, in which each pixel value is
normalized to the interval <img id="Sx1.SSx2.p1.m2" class="ltx_Math" style="vertical-align:-6px" src="mi/mi13.png" width="39" height="21" alt="\left[0,1\right]"> and used to represent
the weighting factor. The average number of molecules at a given integer
pixel position <img id="Sx1.SSx2.p1.m3" class="ltx_Math" style="vertical-align:-6px" src="mi/mi12.png" width="45" height="21" alt="\left(x,y\right)"> is then computed as</p>
<table id="Sx1.Ex1" class="ltx_equation">

<tr class="ltx_equation ltx_align_baseline">
<td class="ltx_eqn_pad"></td>
<td class="ltx_align_center"><img id="Sx1.Ex1.m1" class="ltx_Math" style="vertical-align:-6px" src="mi/mi8.png" width="207" height="24" alt="\overline{d}\left(x,y\right)=a^{2}M\left(x,y\right)d_{\mathrm{max}}\,,"></td>
<td class="ltx_eqn_pad"></td>
</tr>
</table>
<p class="ltx_p">where <img id="Sx1.SSx2.p1.m4" class="ltx_Math" style="vertical-align:-2px" src="mi/mi15.png" width="13" height="12" alt="a"> is the camera pixel size as projected to the sample plane,
and <img id="Sx1.SSx2.p1.m5" class="ltx_Math" style="vertical-align:-5px" src="mi/mi16.png" width="40" height="19" alt="d_{\mathrm{max}}"> is the maximum spatial density of molecules
per unit area as specified by users.</p>
</div>
<div id="Sx1.SSx2.p2" class="ltx_para">
<p class="ltx_p">The procedure for generating molecular positions follows the spatial
Poisson point process <cite class="ltx_cite">[<a href="#bib.bib18" title="The Advanced Theory of Statistics" class="ltx_ref">2</a>]</cite>. Thus for each value <img id="Sx1.SSx2.p2.m1" class="ltx_Math" style="vertical-align:-6px" src="mi/mi14.png" width="57" height="24" alt="\overline{d}\left(x,y\right)">,
a random number of events (molecules) is created in the pixel <img id="Sx1.SSx2.p2.m2" class="ltx_Math" style="vertical-align:-6px" src="mi/mi12.png" width="45" height="21" alt="\left(x,y\right)">
using a Poisson random number generator. Molecular centers are placed
uniformly and randomly within that pixel.</p>
</div>
<div id="Sx1.SSx2.p3" class="ltx_para">
<p class="ltx_p">Note that the mask <img id="Sx1.SSx2.p3.m1" class="ltx_Math" style="vertical-align:-2px" src="mi/mi10.png" width="23" height="16" alt="M"> should be at least the same size as the desired
super-resolution image in order to preserve high resolution in the
final reconstruction. The coordinates of the molecular centers are
down-scaled appropriately.</p>
</div>
</div>
<div id="Sx1.SSx3" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">Additional sample drift</h2>

<div id="Sx1.SSx3.p1" class="ltx_para">
<p class="ltx_p">The generated molecular positions in the image sequence can be subjected
to a lateral sample drift. Users need to specify the speed and direction
of the drift, which is constant throughout the image sequence.</p>
</div>
</div>
<div id="Sx1.SSx4" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">See also</h2>

<div id="Sx1.SSx4.p1" class="ltx_para">
<ul id="I1" class="ltx_itemize">
<li id="I1.i1" class="ltx_item" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">•</span> 
<div id="I1.i1.p1" class="ltx_para">
<p class="ltx_p"><a href="CameraSetupPlugIn.html" title="" class="ltx_ref">Camera setup</a></p>
</div>
</li>
<li id="I1.i2" class="ltx_item" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">•</span> 
<div id="I1.i2.p1" class="ltx_para">
<p class="ltx_p"><a href="PerformanceEvaluationPlugIn.html" title="" class="ltx_ref">Performance evaluation</a></p>
</div>
</li>
</ul>
</div>
</div>
</div>
<div id="bib" class="ltx_bibliography">
<h1 class="ltx_title ltx_title_bibliography">References</h1>

<ul id="L1" class="ltx_biblist">
<li id="bib.bib11" class="ltx_bibitem ltx_bib_article">
<span class="ltx_bibtag ltx_bib_key ltx_role_refnum">[1]</span>
<span class="ltx_bibblock"><span class="ltx_text ltx_bib_author">M. Hirsch, R. J. Wareham, M. L. Martin-Fernandez, M. P. Hobson and D. J. Rolfe</span><span class="ltx_text ltx_bib_year">(2013)</span>
</span>
<span class="ltx_bibblock"><span class="ltx_text ltx_bib_title">A stochastic model for electron multiplication charge-coupled devices–from theory to practice.</span>,
</span>
<span class="ltx_bibblock"><span class="ltx_text ltx_bib_journal">PLoS ONE</span> <span class="ltx_text ltx_bib_volume">8</span> (<span class="ltx_text ltx_bib_number">1</span>), <span class="ltx_text ltx_bib_pages"> pp. e53671</span>.
</span>
<span class="ltx_bibblock">External Links: <span class="ltx_text ltx_bib_links"><a href="http://dx.doi.org/10.1371/journal.pone.0053671" title="" class="ltx_ref doi ltx_bib_external">Document</a></span>.
</span>
<span class="ltx_bibblock ltx_bib_cited">Cited by: <a href="#Sx1.SSx1.p2" title="Image formation ‣ Generator of simulated data" class="ltx_ref"><span class="ltx_text ltx_ref_title">Image formation</span></a>.
</span>
</li>
<li id="bib.bib18" class="ltx_bibitem ltx_bib_book">
<span class="ltx_bibtag ltx_bib_key ltx_role_refnum">[2]</span>
<span class="ltx_bibblock"><span class="ltx_text ltx_bib_author">M. Kendall and A. Stuart</span><span class="ltx_text ltx_bib_year">(1979)</span>
</span>
<span class="ltx_bibblock"><span class="ltx_text ltx_bib_title">The Advanced Theory of Statistics</span>,
</span>
<span class="ltx_bibblock"> <span class="ltx_text ltx_bib_publisher">London: Charles Griffin</span>.
</span>
<span class="ltx_bibblock ltx_bib_cited">Cited by: <a href="#Sx1.SSx2.p2" title="Fixed or spatially varying density of molecules ‣ Generator of simulated data" class="ltx_ref"><span class="ltx_text ltx_ref_title">Fixed or spatially varying density of molecules</span></a>.
</span>
</li>
<li id="bib.bib17" class="ltx_bibitem ltx_bib_article">
<span class="ltx_bibtag ltx_bib_key ltx_role_refnum">[3]</span>
<span class="ltx_bibblock"><span class="ltx_text ltx_bib_author">P. Křížek, I. Raška and G. M. Hagen</span><span class="ltx_text ltx_bib_year">(2011)</span>
</span>
<span class="ltx_bibblock"><span class="ltx_text ltx_bib_title">Minimizing detection errors in single molecule localization microscopy</span>,
</span>
<span class="ltx_bibblock"><span class="ltx_text ltx_bib_journal">Optics Express</span> <span class="ltx_text ltx_bib_volume">19</span> (<span class="ltx_text ltx_bib_number">4</span>), <span class="ltx_text ltx_bib_pages"> pp. 3226–35</span>.
</span>
<span class="ltx_bibblock">External Links: <span class="ltx_text ltx_bib_links"><a href="http://dx.doi.org/10.1364/OE.19.003226" title="" class="ltx_ref doi ltx_bib_external">Document</a></span>.
</span>
<span class="ltx_bibblock ltx_bib_cited">Cited by: <a href="#Sx1.p1" title="Generator of simulated data" class="ltx_ref"><span class="ltx_text ltx_ref_title">Generator of simulated data</span></a>.
</span>
</li>
<li id="bib.bib38" class="ltx_bibitem ltx_bib_article">
<span class="ltx_bibtag ltx_bib_key ltx_role_refnum">[4]</span>
<span class="ltx_bibblock"><span class="ltx_text ltx_bib_author">S. Wolter, M. Schüttpelz, M. Tscherepanow, S. van de Linde, M. Heilemann and M. Sauer</span><span class="ltx_text ltx_bib_year">(2010)</span>
</span>
<span class="ltx_bibblock"><span class="ltx_text ltx_bib_title">Real-time computation of subdiffraction-resolution fluorescence images</span>,
</span>
<span class="ltx_bibblock"><span class="ltx_text ltx_bib_journal">Journal of Microscopy</span> <span class="ltx_text ltx_bib_volume">237</span> (<span class="ltx_text ltx_bib_number">1</span>), <span class="ltx_text ltx_bib_pages"> pp. 12–22</span>.
</span>
<span class="ltx_bibblock">External Links: <span class="ltx_text ltx_bib_links"><a href="http://dx.doi.org/10.1111/j.1365-2818.2009.03287.x" title="" class="ltx_ref doi ltx_bib_external">Document</a></span>.
</span>
<span class="ltx_bibblock ltx_bib_cited">Cited by: <a href="#Sx1.p1" title="Generator of simulated data" class="ltx_ref"><span class="ltx_text ltx_ref_title">Generator of simulated data</span></a>.
</span>
</li>
</ul>
</div>
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